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Extremes

, Volume 20, Issue 3, pp 493–517 | Cite as

Generalized Pickands constants and stationary max-stable processes

  • Krzysztof Dȩbicki
  • Sebastian Engelke
  • Enkelejd Hashorva
Article

Abstract

Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (Bernoulli, 20(3), 1600–1619, 2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker–Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore develop a link to extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

Keywords

Brown–Resnick process Fractional Brownian motion Gaussian process Generalized Pickands constant Lévy process Max-stable process Mixed moving maxima representation 

AMS 2000 Subject Classifications

60G15 60G70 

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Notes

Acknowledgments

We are grateful to three anonymous referees, Thomas Mikosch and Ilya Molchanov for numerous important suggestions. In particular, the new M3 representation (??) was suggested by one of the referees. Financial support by the Swiss National Science Foundation grants 200021-166274 (EH) and 161297 (SE), and partial support by NCN Grant No 2015/17/B/ST1/01102 (2016-2019) (KD) is gratefully acknowledged.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Krzysztof Dȩbicki
    • 1
  • Sebastian Engelke
    • 2
  • Enkelejd Hashorva
    • 3
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.University of LausanneLausanneSwitzerland

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